When you have a compound subject connected with 'and', always use the plural form of the verb.

When you have a compound subject connected with 'or', match the verb to the closest noun.

The sentence "There are progress and improvements" is in passive voice; the "correct" way to say it would be (although few would use it in the real world), "Progress and improvements are there." Or, "Improvements and progress are there."

I would so not write that sentence, but if it were necessary to have it composed in that way, I would say "there is." Of course, that's because, if I'm forced into an awkward sentence, I want the reader to move over it as quickly as possible, and I perceive "there is" would cause less stumbling, not because I'm convinced it's grammatically exact. I can't find the reference right now, but I was looking this up recently and found that the rule is tending toward making the verb match the closer word. "There are improvements and progress" versus "There is progress and improvements."

Some examples, perhaps more obvious, that we expect "There is":

There is such love and joy in my heart.

There is enough time and money to go on vacation.

There is cheese and meat for sandwiches.

The source I was looking at said these were becoming correct because that's what we're coming to expect. Officially, the construction is still "informal," which brings me back to the idea that if that sentence is to be used in formal writing, I just would not use that construction. Others have made useful suggestions for the restructuring that I'll not repeat.